Question

If \(\mathrm{yx}<1\), then: (A) Both \(x\) and \(y\) are negative (B) Both \(x\) and \(y\) are positive (C) Only one of the two is negative (D) \(x

Short answer

The correct answer is (E) None of the above.

Step-by-step solution

Identify the Given Inequality

The inequality given is \(\textstyle{yx < 1}\). This is the foundation of the problem.

Test Values for Different Scenarios - Both Positive

Assume both \(x\) and \(y\) are positive. For instance, let \(x = 0.5\) and \(y = 0.5\). Calculate \(xy = 0.5 \times 0.5 = 0.25 < 1\). This works. Therefore, it is possible that both \(x\) and \(y\) are positive.

Test Values for Different Scenarios - Both Negative

Assume both \(x\) and \(y\) are negative. For example, let \(x = -0.5\) and \(y = -0.5\). Calculate \(xy = -0.5 \times -0.5 = 0.25 < 1\). This also works. Therefore, it is also possible that both \(x\) and \(y\) are negative.

Test Values for Different Scenarios - One Positive and One Negative

Assume one value is positive and the other is negative. For instance, let \(x = 2\) and \(y = -0.5\). Calculate \(xy = 2 \times (-0.5) = -1 < 1\). This works as well. Therefore, it is also possible that one of the numbers is positive and the other is negative.

Consider the Given Answers

Now look at the provided choices: (A) Both \(x\) and \(y\) are negative. (B) Both \(x\) and \(y\) are positive. (C) Only one of the two is negative. (D) \(x < y\). (E) None of the above.

Draw Conclusion

Since \(\textstyle{xy < 1}\) holds true for multiple scenarios (both positive, both negative, and one positive and one negative), none of the given choices fully satisfy all possible cases. Therefore, the correct answer is (E) None of the above.

Key concepts

Understanding Inequalities

Inequalities compare two values, showing if one is less than, greater than, or equal to another. In the given problem, the inequality is represented as \(\textstyle{yx < 1}\). The key is to analyze how different values of \(x\) and \(y\) can satisfy this inequality. We test them in different scenarios: both positive, both negative, and one positive and one negative. Understanding these comparisons helps in determining the broad range in which solutions can exist. To effectively handle inequalities, always remember to test various values and combinations to ensure all potential solutions are explored.

Testing Scenarios

Testing multiple scenarios is crucial for solving inequalities like \(\textstyle{yx < 1}\). We break down the process into specific cases:

• Both \(x\) and \(y\) being positive: For example, \(x = 0.5\) and \(y = 0.5\), yielding \(0.25 < 1\).
• Both \(x\) and \(y\) being negative: Taking \(x = -0.5\) and \(y = -0.5\), which gives \(0.25 < 1\).
• One positive and one negative. For instance, \(x = 2\) and \(y = -0.5\), yielding \(-1 < 1\).

This methodical approach verifies or refutes potential answers, ensuring a thorough examination of the inequality under various conditions.

Logical Reasoning

Logical reasoning underpins the whole problem-solving process, helping to draw accurate conclusions from given data. Here, the statement \(\textstyle{yx < 1}\) sets the stage. By systematically testing different combinations of \(x\) and \(y\) values, we use logical reasoning to conclude:

• Both numbers can be positive or negative.
• One can be positive while the other is negative.
  

These steps inform us that the given answer choices don't fully satisfy the inequality under all circumstances, leading to answer choice (E): None of the above. Logical reasoning ensures each step is grounded in sound analysis.

Mathematical Proofs

Mathematical proofs validate our conclusions using logical steps and evidence. We start with the inequality \(\textstyle{yx < 1}\) and explore different combinations of \(x\) and \(y\). Each scenario is a mini-proof:

• Positive values: \(x = 0.5\), \(y = 0.5\) → \(0.25 < 1\).
• Negative values: \(x = -0.5\), \(y = -0.5\) → \(0.25 < 1\).
• Mixed values: \(x = 2\), \(y = -0.5\) → \(-1 < 1\).

These proofs collectively show \(\textstyle{yx < 1}\) holds true across different scenarios, confirming that none of the answer choices (A-D) satisfy all conditions. Hence, the correct answer is (E).